Abstract: Conditions are formulated which guarantee the existence of positive solutions for systems of the form

, where is the Laplacian (with Dirichlet boundary conditions) on an open domain in , and where each is a positive measure. The main tools used are probabilistic potential theory, Markov processes, and an iterative scheme which is not a generalization of the one used for quasimonotone systems. Quasimonotonicity is not assumed and new results are obtained even for the case where for every and .